V. Vu has recently shown that when k≥2 and s is
sufficiently large in terms of k, then there exists a set $\mathfrak{X} (k), whose
number of elements up to t is smaller than a constant
times (t log t)1/s, for which all
large integers n are
represented as the sum of s kth powers of elements of
$\mathfrak{X} (k) in order
log n ways. We establish this conclusion with
s∼k log k, improving on the constraint implicit in Vu's work
which forces s to be as large as k48k. Indeed, the
methods of this paper show, roughly speaking, that whenever existing
methods permit one to show that all large integers are the sum of
H(k) kth powers of natural numbers, then H(k)+2 variables suffice
to obtain a corresponding conclusion for "thin sets," in the sense of
Vu.